Numerical Calculations on Lithium-ion Batteries

Group 16By:

Albert Ho, Eric Vavra, Mitchell Gee, Justin Matson, Michelle Empleo

http://periodictable.com/Elements/003/http://batt.lbl.gov/blog/research-tasks/insights-into-designing-faster-charging-batteries/

http://www.extremetech.com/computing/153614-new-lithium-ion-battery-design-thats-2000-times-more-powerful-recharges-1000-times-faster

http://philosophyofscienceportal.blogspot.com/2010/06/deadly-lithium-batteries.html

http://www.digitaltrends.com/cool-tech/scientists-develop-way-to-triple-battery-life-in-electronics/

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Introduction: ChargingLithium ion batteries are a type of rechargeable battery that is charged

by lithium ions moving from the cathode to the anode when charging and from anode to cathode when discharging.

2http://www.scielo.br/scielo.php?pid=s0103-50532006000400002&script=sci_arttext

Introduction: ApplicationLithium-Ion batteries are common in portable electronic devices. These

batteries are also being used in some electric cars as well as numerous military purposes. They are known for having a good balance of energy density and power density.

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http://www.maxxia.com.au/salary-packaging/what-can-i-salary-package/portable-electronic-devices-for-work

http://newpard.en.busytrade.com/products/info/2119308/Fenix-Tk15-Flashlight-Use-By-Cr123a-Li-ion-Batteries.html

Introduction: Energy DensityEnergy density is the amount of energy that a battery can store, which is

decided by the chemistry of the cell. Lithium ion batteries are becoming very popular for having higher energy density than other batteries with around 115 Wh/kg.

4http://inventorspot.com/articles/lithium_ion_batteries_future_power_and_portability

Introduction: Power DensityPower density is the rate at which energy can be delivered from the

battery to the device. The amount of this is determined by the cell design and kinetics. Lithium ion’s good balance between energy density and power density can be seen on the chart below.

5http://electronics.howstuffworks.com/everyday-tech/lithium-ion-battery.htm

http://www.extremetech.com/computing/153614-new-lithium-ion-battery-design-thats-2000-times-more-powerful-recharges-1000-times-faster

Method 1: Gibb’s Free EnergyA major challenge in battery development is to maximize the energy density. Energy density is a

major concern in cell phones. The energy density of a battery is the product of the cell’s voltage and specific capacity. The voltage corresponding to a given-transfer reaction can be related to the Gibb’s free energy.

Gibb’s Free Energy is a thermodynamic potential that measures the “usefulness” or process-initiating work obtainable from a thermodynamic system at constant temperature and volume.

Gibb’s Free Energy can be determined from the equation: ΔG = -nFE° ΔG = Gibb’s Free Energyn = Number of electrons involved in the reaction = 2 e -

F = Faraday’s constant = 96487 coulomb/mole (C/mol)E° = Cell Voltage (V)

http://upload.wikimedia.org/wikibooks/en/a/a6/Gibbs_free_energy.JPG

http://wikis.lawrence.edu/display/CHEM/Free+Energy+-+Shank

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Table 1This is a table that we took from our article. It is basically just a list of different electrochemical materials that

are used in battery electrodes (referenced to a hydrogen electrode. The reactions of each element and

their respective cell voltages.We excluded the oxygen and lithium parts of this table because we

wanted to hold n as a constant of 2 electrons and these two elements and 4 and 2 electrons involved in

the reaction respectively.

http://0.tqn.com/d/inventors/1/0/e/H/battery1.gif7Spotnitz, Robert, “Lithium-Ion Batteries: The Basics”,

Journal, AIChE, October, 2013, online, 11/20/2013

ResultsElement used ΔG, Gibb’s Free Energy

F2, Fluorine -553835

Cl2, Chlorine -262445

PbO2, Lead Oxide -325161

Ag, Silver -65997.1

H2, Hydrogen 0

Pb, Lead 25086.6

Cd, Cadmium 82978.8

Zn, Zinc 146660

Zn(OH)2, Zinc Hydroxide 247972 8

Numerical Optimization of Gibb’s Free Energy

Direct plotting of ΔG vs. E° cell with constant n allows us to find the local minimum value for ΔG analytically and thus to find the most spontaneous cell.

The MATLAB script is written as follows:

%create values of x in a vector

x=[2.87, 1.36, 1.685, 0.342, 0, -0.13, -.43, -.76, -1.285]

%function for Gibb's free energy change defined:

y=-2*96487.*x

%create plot of y vs. x

plot(x,y,'o')

hold on

plot(x,y)

xlabel ('Cell Voltage (V)')

ylabel ('Free Energy (C*V/mol)')

title ('Free Energy vs. Voltage')

grid

hold off

http://www.mathsisfun.com/algebra/functions-maxima-minima.html

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.m Script and Graph

From the plot, we can observe a minimum value of -553825 (C*V/mol) for ΔG occurs at a value of -3.05 for E° cell.

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Optimization in MATLAB Cont.We can confirm the answer from the previous slide using MATLAB’s

fminbnd function • fminbnd uses the golden section search

as well as parabolic interpolation to find the

minimum value of our function on our given

interval

We can employ the following MATLAB

script to accomplish this task:

f=@(x) -2*96487*x;

format longg

[x,min]=fminbnd(f,-3.05,2.7)

http://www.mathsisfun.com/numbers/golden-ratio.html

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fminbnd Results

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Method 2: Mass- Transfer- LimitedCurrent Density, id

Two main categories limit the power of batteries:

1. Voltage loss associated with charge transport2. Mass- Transfer- Limited Current Density

http://newsavalanche.com/2013/04/09/tin-nanocrystals-form-future-battery-how-li-ion-works/http://batt.lbl.gov/blog/research-tasks/insights-into-desig

ning-faster-charging-batteries/ 13

Current Density DefinitionThe effect of mass transport on the charge (or discharge) rateEquation:

id = mass transfer limited current density (A/m2)

Deff = effective diffusion coefficient (m2/s)

c = concentration of lithium salt in the electrolyte (mol/m3)

δ = diffusion length (m)

http://www.nature.com/srep/2013/130605/srep01946/full/srep01946.html

14http://www.nature.com/srep/2013/130605/srep01946/images_article/srep01946-f1.jpg

Linear Regression

Solid-phase diffusion coefficients (Deff) are typically about 10-13 m2/s

We can use the numerical method linear regression in order to find a value for Deff

for a lithium-ion battery given a table with the current densities vs. the concentration of lithium:

http://www.prlog.org/11488430-breaking-world-record-on-charger-by-using-lithium-ion-battery.html

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Linear Regression Calculations Given the info for a monolithic nonporous electrode:

F = Faraday’s Constant (96,487 C/mol)δ = 30 * 10-6 m

Since F and δ are constants, we can plot id vs. c in order to find the slope. With the slope, we can find Deff. This can be done in matlab with the program on the next slide.

We can calculate Deff:

Spotnitz, Robert, “Lithium-Ion Batteries: The Basics”, Journal, AIChE, October, 2013, online, 11/20/2013

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Matlab Programfunction [s, r] = regression(c, id)

% input:% c = concentration (mol/m^3)% id = mass transfer limited current density (A/m^2)% output:% s = vector of the slope s(1) and intercept s(2)% r = coefficient of determination

n = length(c);if length(id) ~=n, error(‘c and id must be same length’); end

c = c(:); id= id(:);sc = sum(c); sid = sum(id);sc2 = sum(c.*c); scid = sum(c.*id); sid2 = sum(id.*id);

s(1) = (n*scid-sc*sid)/(n*sc2-sc^2);s(2) = sid/n-s(1)*sc/n;

r = ((n*scid-sc*sid)/sqrt(n*sc2-sc^2)/sqrt(n*sid2-sid^2))^2;

http://www.telegraph.co.uk/education/9600921/Computer-programming-who-is-teaching-our-children-to-code.html

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Matlab Program cont.% Plot of the data and best fit line

cp = linspace(min(c),max(c),2);idp = s(1)*cp+s(2);plot(c,id,’o’,cp,idp)grid on

http://my.bestfitlineruler.com/index.php/instructions-for-use/step-4/ 18

Matlab Calculations

Using the matlab function, the slope of the line is 0.32348 * 10-3. Plugging this slope into the formula gives you a Deff of 1.0057 * 10-13 m2/sec, which is close to the approximate value of 10-13 m2/s.

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Conclusion • Numerical methods can be applied

to find the Gibb’s free energy of differentelements used in batteries.

• Determining the Gibb’s free energy value can tell us which element(s) are most efficient when creating the optimal performance battery.

• With the data found we determined that the zinc hydroxide would perform the best based on its Gibb’s free energy change

http://img97.imageshack.us/img97/1782/12345lu.jpg

http://fineartamerica.com/featured/zinc-hydroxide-precipitate-andrew-lambert-photography.html

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Conclusion (cont.)• We can use numerical methods

in order to find a value for Deff for a

lithium-ion battery given a table with the current densities vs. the concentration of lithium

• We can use the Deff to determine

what diffusion coefficient produces the highest current density amongst lithium ion batteries.

http://www.houseofbatteries.com/images/Comparison_chart.jpg

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Future WorkSeparators • Separators prevent electronic contact

between electrodes while allowing ionic transport

• nanofiber-based polymeric battery separator• Developed by DuPont

• Increases power up to 30 %

• Increases battery life up to 20 %

• Added thermal stability

http://evworld.com/press/dupont_hybrid_ghostillust.jpg

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Future WorksSupercapacitors

Supercapacitors are capacitors which have low energy density. The organic electrolyte used in supercapacitors allows for fast energy discharge of which is more rapid than that of a battery.

• A team of researchers at UCLA discovered a way to create graphene-based supercapacitors.

• Operate three times faster than lithium batteries

A Maxwell Technologies supercapacitor cell and two different

multi-cell modules http://commons.wikimedia.org/

UCLA researchers develop new technique to scale up production of graphene micro-supercapacitors. http://newsroom.ucla.edu/portal/ucla/ucla-researchers-develop-new-technique-243553.aspx

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Future WorksThe lithium-air battery

• Oxidizes lithium at the anode and reduces oxygen at the cathode which causes current flow.

• High energy density

• Energy density (per kilo) comparable to the energy density of gasoline per kilo.

• Do not store an oxidizer internally since oxygen is used from air

http://images.gizmag.com/hero/lithium-air-battery.jpg

http://insideevs.com/wp-content/uploads/2013/09/li-air.jpg

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Future Works Liquid-Based Batteries

• Most traditional battery research has focused on solid-state chemistry and physics principles that have been used for the past two centuries

• Liquid batteries remove some of the conventional restrictions associated with solid-state batteries.

• You can store charge in a liquid that you can pump through the battery and then you can recharge the battery by reversing the flow of the pump

http://onlinelibrary.wiley.com/store/10.1002/cssc.201200730/asset/image_m/mcontent.gif?v=1&s=db2f5d12c8020107daddec0002810d18d62deb0d

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Works CitedArticles:1 Spotnitz, Robert, “Lithium-Ion Batteries: The Basics”, Journal, AIChE, October, 2013, online, 11/20/2013

2 Pikul, James H., et. al., “High-power lithium ion microbatteries from interdigitated three-dimensional bicontinuous nanoporous electrodes”, Nat Commun. 2013/04/16/online 11/23/2013

3 Reddy,T.,ed., “Linden’s Handbook of Batteries,” 4th ed., McGraw-Hill, New York, NY (2011) online, 11/23/2013

4 Newman, J., and K.E. Thomas-Alyea, “Electrochemical Systems,” 3rd ed., Wiley, Hoboken, NJ (2004) online, 11/23/2013

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http://batt.lbl.gov/blog/research-tasks/insights-into-designing-faster-charging-batteries/

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http://iwagemusic.com/digital-library-update/